Wordpress - Programmatically add a Navigation menu and menu items

I might be misunderstanding you, but why not use wp_create_nav_menu()?

E.g., this is what I do to create a custom BuddyPress menu when I detect BP as active:

    $menuname = $lblg_themename . ' BuddyPress Menu';
$bpmenulocation = 'lblgbpmenu';
// Does the menu exist already?
$menu_exists = wp_get_nav_menu_object( $menuname );

// If it doesn't exist, let's create it.
if( !$menu_exists){
    $menu_id = wp_create_nav_menu($menuname);

    // Set up default BuddyPress links and add them to the menu.
    wp_update_nav_menu_item($menu_id, 0, array(
        'menu-item-title' =>  __('Home'),
        'menu-item-classes' => 'home',
        'menu-item-url' => home_url( '/' ), 
        'menu-item-status' => 'publish'));

    wp_update_nav_menu_item($menu_id, 0, array(
        'menu-item-title' =>  __('Activity'),
        'menu-item-classes' => 'activity',
        'menu-item-url' => home_url( '/activity/' ), 
        'menu-item-status' => 'publish'));

    wp_update_nav_menu_item($menu_id, 0, array(
        'menu-item-title' =>  __('Members'),
        'menu-item-classes' => 'members',
        'menu-item-url' => home_url( '/members/' ), 
        'menu-item-status' => 'publish'));

    wp_update_nav_menu_item($menu_id, 0, array(
        'menu-item-title' =>  __('Groups'),
        'menu-item-classes' => 'groups',
        'menu-item-url' => home_url( '/groups/' ), 
        'menu-item-status' => 'publish'));

    wp_update_nav_menu_item($menu_id, 0, array(
        'menu-item-title' =>  __('Forums'),
        'menu-item-classes' => 'forums',
        'menu-item-url' => home_url( '/forums/' ), 
        'menu-item-status' => 'publish'));

    // Grab the theme locations and assign our newly-created menu
    // to the BuddyPress menu location.
    if( !has_nav_menu( $bpmenulocation ) ){
        $locations = get_theme_mod('nav_menu_locations');
        $locations[$bpmenulocation] = $menu_id;
        set_theme_mod( 'nav_menu_locations', $locations );
    }

As a complement to ZaMoose's anwser, here's how you would create a "Page-type" menu item (not a "Custom" one):

wp_update_nav_menu_item($menu_id, 0, array('menu-item-title' => 'About',
                                           'menu-item-object' => 'page',
                                           'menu-item-object-id' => get_page_by_path('about')->ID,
                                           'menu-item-type' => 'post_type',
                                           'menu-item-status' => 'publish'));

Assuming you only know the page slug, for instance.

I have a few issues with the accepted answer - that doesn't make it wrong, but I'll post my own code below which I feel might have a better result for some people since I had the same question but wanted to do the same thing with less code.

First, the above code creates "URL" type navigation items, which is fine for some people but I want to link to PAGES, not URLs because this is an important feature of WordPress navigations and clients inevitable move things around so I never use the URL nav item type.

Also, only a flat array of children is handled by the posted code. I've created a function for recursively declaring the new nav items, storing their returned metadata (mainly ID after being created in the loop), and a parameter for accepting children.

Just edit $nav_items_to_add and the rest is handled recursively. There are 3 required keys in each array. First, the array key is the slug, so 'shop' => array( ... ) is what you want for a page with the slug shop. ['title']is the way the nav item will be labeled on the front end. path is the path to the page within the WordPress page hierarchy, so this is identical to the slug if the page is a top-level parent, and if shop were a child of home then it would be 'path' => 'home/shop'.

The last optional array key is ['parent'] where you can declare another key in the array as the parent of the current one. It's important to note that the items are added recursively, so the parent has to exist before you try to create a child. This means the declaration should happen for the parent nav item before it's children.

    $locations = get_nav_menu_locations();

    if (isset($locations['primary_navigation'])) {
        $menu_id = $locations['primary_navigation'];

        $new_menu_obj = array();

        $nav_items_to_add = array(
                'shop' => array(
                    'title' => 'Shop',
                    'path' => 'shop',
                    ),
                'shop_l2' => array(
                    'title' => 'Shop',
                    'path' => 'shop',
                    'parent' => 'shop',
                    ),
                'cart' => array(
                    'title' => 'Cart',
                    'path' => 'shop/cart',
                    'parent' => 'shop',
                    ),
                'checkout' => array(
                    'title' => 'Checkout',
                    'path' => 'shop/checkout',
                    'parent' => 'shop',
                    ),
                'my-account' => array(
                    'title' => 'My Account',
                    'path' => 'shop/my-account',
                    'parent' => 'shop',
                    ),
                'lost-password' => array(
                    'title' => 'Lost Password',
                    'path' => 'shop/my-account/lost-password',
                    'parent' => 'my-account',
                    ),
                'edit-address' => array(
                    'title' => 'Edit My Address',
                    'path' => 'shop/my-account/edit-address',
                    'parent' => 'my-account',
                    ),
            );

    foreach ( $nav_items_to_add as $slug => $nav_item ) {
        $new_menu_obj[$slug] = array();
        if ( array_key_exists( 'parent', $nav_item ) )
            $new_menu_obj[$slug]['parent'] = $nav_item['parent'];
        $new_menu_obj[$slug]['id'] = wp_update_nav_menu_item($menu_id, 0,  array(
                'menu-item-title' => $nav_item['title'],
                'menu-item-object' => 'page',
                'menu-item-parent-id' => $new_menu_obj[ $nav_item['parent'] ]['id'],
                'menu-item-object-id' => get_page_by_path( $nav_item['path'] )->ID,
                'menu-item-type' => 'post_type',
                'menu-item-status' => 'publish')
        );
    }

    }